Local classfield theory

In mathematics, local classfield theory is the study in number theory of the abelian extensions of local fields. It is in itself a rather successful theory, leading to definite conclusions. It is also important for (and was developed to help elucidate) the proofs of class field theory itself.

Related Topics:
Mathematics - Number theory - Abelian extension - Local field - Class field theory

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The basic theory concerns for a local field K, the description of the Galois group G of the maximal abelian extension of K. This is closely related to K×, the multiplicative group of K{0}. These groups cannot be equal: as a topological group G is pro-finite and so compact. On the other hand K× is not compact.

Related Topics:
Galois group - Multiplicative group - Pro-finite - Compact

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Taking the case where K is a finite extension of the p-adic numbers Qp, we can say more precisely that K× has the structure of a cartesian product of a compact group with an infinite cyclic group. The main topological operation is to replace the infinite cyclic group by a group Z^, i.e. its pro-finite completion with respect to subgroups of finite index. This can be done by indicating a topology on K×, for which we can complete. This, roughly speaking, is then the correct group to identify with G.

Related Topics:
Finite extension - P-adic number - Cartesian product - Infinite cyclic

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The actual isomorphism used is important in practice, and is described in the theory of the norm residue symbol.

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